Rate of change question?

Question

I do not know how to do this problem. Please help.

The top of a 17-foot ladder is sliding down a vertical wall at a constant rate of 2 feet per minute. When the top of the ladder is 8 feet from the ground, what is the rate of change of the distance between the bottom of the ladder and the wall?

Thanks!

Answer 1

Let:

y be the vertical height of the top of the ladder,

x be the distance of the foot of the ladder from the wall.

x^2 + y^2 = 17^2 ...(1)

Differentiating (1) with respect to time:

2x(dx / dt) + 2y(dy / dt) = 0

Substituting dy / dt = - 2 ft/min:

2x(dx / dt) - 4y = 0

dx / dt = (4y) / (2x)

= 2y / x

When y = 8 ft:

dx / dt = 16 / x ...(2)

(1) gives:

x = sqrt(17^2 - 8^2)

= 15.

Substituting for x in (2):

dx / dt = 16 / 15 ft/min.